aa r X i v : . [ h e p - t h ] S e p Soft Hair as a Soft Wig
Raphael Bousso a and Massimo Porrati b a Center for Theoretical Physics and Department of Physics, University of California,Berkeley, CA 94720, USA and
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA b Center for Cosmology and Particle Physics,Department of Physics, New York University,726 Broadway, New York, NY 10003, USA Abstract
We consider large gauge transformations of gravity and electromagnetism in D = 4asymptotically flat spacetime. Already at the classical level, we identify a canonical trans-formation that decouples the soft variables from the hard dynamics. We find that onlythe soft dynamics is constrained by BMS or large U (1) charge conservation. Physicallythis corresponds to the fact that sufficiently long-wavelength photons or gravitons that areadded to the in-state will simply pass through the interaction region; they scatter triviallyin their own sector. This implies in particular that the large gauge symmetries bear norelevance to the black hole information paradox. We also present the quantum version ofsoft decoupling. As a consistency check, we show that the apparent mixing of soft andhard modes in the original variables arises entirely from the long range field of the hardcharges, which is fixed by gauge invariance and so contains no additional information. email: [email protected] email: [email protected] Introduction
Large gauge transformations in asymptotically flat spacetime are generated by an infinite set ofcharges, Q [ f ] [1–3]. Here f (Θ) is an arbitrary function on the celestial sphere, so there is oneindependent charge per solid angle Θ. Unlike a pure gauge constraint, the value of Q [ f ] doesnot vanish identically, so it connects distinct states in phase space.The charge consists of a soft and a hard part, Q [ f ] = Q s [ f ] + Q h [ f ], defined in moredetail below. In massless electromagnetism, Q s generates an angle-dependent change of thegauge potential, and Q h generates the compensating phase changes of charged fields. Anotherimportant example is gravity, where Q [ f ] generates an asymptotic diffeomorphism called Bondi-van der Burg-Metzner-Sachs (BMS) supertranslation [4]. In this case, Q h generates an angle-dependent time translation of all radiative modes, and Q s generates a compensating deformationof asymptotic coordinate spheres [5–7].The soft and hard parts are not separately conserved. But with a suitable identificationof the gauge potentials near spatial infinity, the total charge is conserved: Q + [ f ] = Q − [ f ].(Superscripts ± refer to past and future null infinity, I ± .)Thus, it would appear that asymptotic symmetries place interesting constraints on the scat-tering problem, at least in D = 4 dimensions. Indeed, Hawking, Perry and Strominger [8, 9]have speculated that the conservation laws associated with BMS supertranslations or large U (1)transformations may have some bearing on the black hole information paradox [10].The purpose of this note is to point out that this is not the case. We show that the asymptoticsymmetry group is entirely accounted for by the physically obvious freedom to add sufficientlysoft (long-wavelength) particles, which propagate freely through the hard interaction region. Itis clear that soft particles cannot affect hard scattering, for otherwise experiments at the LHCcould not be analyzed without detailed knowledge of the cosmic microwave background.The simplicity of the soft dynamics is somewhat obscured in D = 4, because the softsector inherits some nontrivial dynamics from the hard scattering. Hard particles have a longrange field, and in D = 4 this tail corresponds to excited radiative modes at arbitrarily largewavelength. (This is not the case in D > only nontrivial dynamics in the soft sector. Since it is completelydetermined by the hard process via the gauge constraints, it yields no independent informationabout the hard scattering.To make this explicit, we perform a simple canonical transformation, or “dressing,” on thedynamical variables. The transformation removes the dynamical effects of the hard scattering1n the soft sector. We find that the dynamics of the two sectors factorizes. Our result is closelyrelated to the IR factorization used in [11] to remove soft hair (see also [12–16]). Ref. [11]uses quantum factorization formulas that apply to photons or gravitons with small but nonzerofrequency; whereas the factorization defined here applies directly to the zero-frequency modesthat are at the origin of the infinite degeneracy of vacua. We also find that it applies both atthe classical and at the quantum level.We also show that the BMS and U (1) large gauge charge does vanish, like a pure gaugetransformation, for a class of dressed states. This class is general enough that an arbitrary statecan be reached by adding further soft excitations that are unrelated to the long range field ofthe hard particles. The fact that Q [ f ] need not vanish can thus be understood entirely in termsof our freedom to add these additional unconstrained soft modes, which scatter trivially.In future work we will argue that with standard definitions of Q s and its conjugate, neitherquantity is observable. However, this is an orthogonal problem, and it can be remedied [17].In the present paper, we consider a different question, namely whether large gauge symmetriesnontrivially constrain the scattering problem in D = 4. The answer is no, regardless of whetherthe soft sector is observable.In Section 2, we introduce the asymptotic symmetries and associated conserved charges.Section 3 is the heart of the paper. In Sec. 3.1 we consider the classical case. We perform acanonical transformation and show that the evolution of the new, dressed variables is completelyindependent of the soft-hair state. The soft sector decouples from the hard dynamics andbecomes trivial. In Sec. 3.2, we obtain the quantum mechanical counterpart of this result. Thisis related to the well known infrared factorization property, which guarantees the absence ofsoft infrared divergences in QED or perturbative quantum gravity. In Section 4 we derive thephysical interpretation of the dressing transformation. We show that it subtracts from the softsector the excitations caused by the long range field of the hard particles. Thus, it removes itsonly nontrivial dynamics. Appendix A shows how to dress gravitational memories defined bythe permanent displacement of massive detectors subject to gravitational radiation in a waythat makes factorization of the BMS supertranslation dynamics manifest.We need not include a discussion of the “black hole horizon charges” considered in [8, 9].In any complete quantum treatment, the only well-defined observables reside at I ± , and our Even when the semiclassical approximation is valid, the definition of “horizon charges” is subject to severeambiguities, and we are not aware of a controlled limit where they make sense. If one fixes the radius of aSchwarzschild black hole while taking G → ∞ , then the total hard flux at I + diverges because of the Hawkingradiation [10]. Moreover, the evaporation timescale diverges, so it is not clear what one would mean by softmodes and charges. In this section, we introduce the large gauge transformations of electromagnetism and gravityin four dimensions. We identify associated charges and conservation laws. We will use theformalism developed in [18] and extensively used in [1, 3, 5, 6, 8, 9]. U (1) Symmetry
Large U (1) symmetries are Abelian gauge transformations that do not vanish at null infinity I ± of Minkowski space. Transformation at I + The large U (1) symmetry is generated by a charge [1, 3] defined at I + − , the past boundary of I + Q + [ f ] = 1 e Z I + − f ∗ F = 1 e Z I + df ∧ ∗ F + Z I + f ∗ J − Z I ++ f ∗ F . (1)The last equality uses Maxwell’s equations d ∗ F = e ∗ J . We assume that there are masslesscharges that can reach I + . We also assume that no charge remains inside the spacetime at latetimes, so the last term, defined at the future boundary of I + , is equal to zero.In retarded coordinates ds = − du − du dr + r γ AB d Θ A d Θ B , (2)the charges then take the form Q + [ f ] = Q + h + Q + s ,Q + h [ f ] = Z I + du d Θ √ γ f j u , Q + s [ f ] = 1 e Z I + du d Θ √ γ f D A F uA . (3)Here the covariant derivative D A is defined with respect to the metric γ AB on the unit coordinatesphere, by D A γ BC = 0. Indices A, B are raised and lowered with γ AB .The large gauge transformation preserves the gauge A r = 0 , A u | I + = 0; so the soft charge3 s [ f ] can also be written as Q s [ f ] = e − Z ∞−∞ du Z d Θ √ γ f D A ∂ u A A . (4)The hard charge Q + h is defined in terms of the boundary current j u ≡ lim r →∞ r J u . Physicallythis corresponds to massless charged particles that reach I ± .The soft charge Q + s [ f ] is the “electromagnetic memory” of an infinite time interval. Phys-ically this is not even approximately observable [17]. However, a finite-time memory can begenerated by soft photons. It can be observed as a change in the relative phases of test chargesstationed at large radius. An observable version of Q s [ f ] will be introduced in [17]. The dis-tinction is irrelevant for our present purposes, so we will use the idealization (4). Q s [ f ] has vanishing Poisson brackets with all matter fields and with the radiative modes F uA | I + . To obtain a symplectic structure in the soft sector, one can introduce a mode φ + (Θ)by [1] lim u →−∞ A A ( u, r, Θ) | I + = − ∂ A φ + (Θ) . (5)and impose the Poisson bracket [ Q + s [ f ] , φ + (Θ)] = − if (Θ) . (6)Thus Π + ( ¯Θ) ≡ Q + s [ δ (Θ − ¯Θ)] (7)is the momentum canonically conjugate to φ + (Θ).As defined, φ + , like Q s , is completely unobservable. Again, an alternate definition can begiven in terms of observable soft photon wavepackets [17]. This does not affect our analysishere, so we adopt the conventional definition above.Expansion into spherical harmonics, f (Θ) = P ∞ l =0 P lm = − l f lm Y lm (Θ), yields a countableinfinity of canonically conjugate operators. Upon quantization they become creation and anni-hilation operators, which generate a “soft” Fock space with a countable basis. Once Gauss law constraints are solved and independent dynamical variables have been identified, there isno difference between the Poisson brackets of those variables and the Dirac bracket. ransformation at I − and Matching Near I − the metric is ds = − dv + 2 dvdr + r γ AB d Θ A d Θ B . The angular coordinates on I − are identified with those at I + by requiring that a null geodesic issuing from I − and passingthrough r = 0 ends up at the same value of the angular coordinate. In other words, the anglesΘ here are antipodally identified with those in Eq. (2). The conserved charge is Q − [ f ] = Q − h + Q − s , (8) Q − h [ f ] = Z I − dv d Θ √ γ f j v , Q − s [ f ] = 1 e Z I − dv d Θ √ γ f D A F vA . (9)The mode canonically conjugate to Π − ( ¯Θ) ≡ Q − s [ f ], f (Θ) = δ (Θ − ¯Θ), is φ − (Θ), which isdefined similarly to φ + as lim v → + ∞ A A ( v, r, Θ) | I − = − ∂ A φ − (Θ) . (10)The soft degrees of freedom φ + , Π + on I + are identified with those on I − by imposingmatching conditions. This is a necessary step to define a scattering problem.The first condition is conservation of the total charge, Q + [ f ] = Q − [ f ]. The rationale forimposing this condition comes from studying the electromagnetic fields produced by movingcharges that interact in a finite region of Minkowski space. In physically sensible generic so-lutions [3, 19] the field strength F ru at any angle Θ on I + − equals F rv on I − + at the antipodalangle. By Eq. (1), the total charge can be expressed only in terms of those quantities and sois antipodally conserved at every angle. Expressed as the sum of soft and hard charge, thematching condition becomesΠ + (Θ) + Z I + du √ γ j u ( u, Θ) = Π − (Θ) + Z I − dv √ γ j v ( v, Θ) . (11)This matching condition can be violated by superimposing free gauge fields that do not vanishat v → + ∞ to initial field configurations that do satisfy it. Finite-energy free fields do vanishat v = ∞ , so Eq. (11) may turn out to be equivalent to imposing finite energy, though we arenot aware of a proof.The second matching condition can be chosen to be φ + (Θ) = φ − (Θ) . (12)This choice is invariant under CPT and Lorentz transformations and is consistent with the5symptotic behavior of generic potentials for moving charges [3,19]. Matching conditions (11,12)lead to standard soft photon/graviton theorems in perturbative quantum field theory [1, 5].(However, we are not aware of a relation of (12) to first principles such as boundedness of theenergy.) The matching conditions (11,12) are the essential ingredient for deriving ourfactorization result in Section 3. The triviality of soft hair is a consequence ofEqs. (11,12) and only of them. No input about the dynamics of the interior ofspacetime is required to derive the results of Section 3. The metric of an asymptotically flat space near I + is (see [9] for notation and normalizations) ds = − du − dudr + r γ AB d Θ A d Θ B + rC AB d Θ A d Θ B + ... (13)The Bondi news is N AB = ∂ u C AB .The metric retains the above form under large diffeomorphisms generated by the BMScharge [5, 6, 9] Q + = Q + h + Q + s ,Q h [ f ] = 14 π Z I + du d Θ √ γ f (Θ) T uu , Q + s [ f ] = − πG Z I + du d Θ √ γ f (Θ) D A D B N AB ,T uu = 18 G N AB N AB + lim r →∞ r T Muu , T M = matter stress-energy tensor . (14) Q + s commutes with N AB and matter fields. To complete the symplectic structure, we introducea boundary field C (Θ) vialim u →−∞ C AB ( u, Θ) = − D A D B C (Θ) + γ AB D C (Θ) ; (15)and we impose the Poisson bracket[ Q + s [ f ] , C + (Θ)] = − if (Θ) . (16)Then Π + ( ¯Θ) ≡ Q + s [ δ (Θ − ¯Θ)] (17)6s the momentum canonically conjugate to C + (Θ).One can likewise define an asymptotic metric near I − in terms of angles θ A , radius r andadvanced time v : ds = − dv − dvdr + r γ AB dθ A dθ B + rC − AB dθ A dθ B + .... (18)The past Bondi news is N − AB = ∂ v C − AB . The past charges Q − [ f ] = Q − h [ f ] + Q − s [ f ] and thecanonical pairs C − ( θ ) and Π − ( θ ) at I − are defined in complete analogy with the future chargesat I + .The matching of angular coordinates on I ± can be performed by continuing the generatorsof I + through space-like infinity i . This again corresponds to an antipodal identification ofangles, to the extent that this bulk notion is well-defined (e.g., in the vacuum). After thisidentification we drop the distinction between angular coordinates on I + and I − and we callthem both Θ A , where it is understood that the same label corresponds to antipodal generatorson I ± .We therefore get the following matching for the soft degrees of freedom:Π + (Θ) + η + (Θ) = Π − (Θ) + η − (Θ) , C + (Θ) = C − (Θ) ,η + (Θ) ≡ π Z I + du √ γ T uu ( u, Θ) , η − (Θ) ≡ π Z I − dv √ γ T vv ( v, Θ) . (19)Apart from obvious changes of names these are the same conditions that we encountered in the U (1) problem. The relation between past in variables at I − and future out variables at I + defines a classicalscattering problem. We can solve this problem directly in the BMS case without passing throughthe “warmup” case of large U (1) symmetry, since the formalism is identical for both cases.In classical mechanics the relation between past and future is given by a symplectic transfor-mation, which is itself defined by a generating functional [20]. The generating functional can betaken to depend on the initial coordinates C − , . . . and final momenta Π + , . . . , where . . . standsfor all other canonical coordinates. 7o define a generating functional one must also partition the hard Bondi news into symplecticsets of commuting variables. This can be done in many ways. A convenient one, tailored tomake contact with the definition of creation and annihilation operators for gravitons, is to definethe positive-frequency part of the news N AB ( u, Θ) as “coordinates” and the negative-frequencyones as “momenta.” (These variables are independent of the boundary variables C ± , unlike the C ± AB .)The form of the matching conditions (19) suggests that we perform the following canonicaltransformation on the in and out variables:Π + (Θ) → Π + D (Θ) = Π + (Θ) + η + (Θ) ,N AB ( u, Θ) → N DAB ( u, Θ) = N AB ( u − C + (Θ) , Θ) , Π − (Θ) → Π − D (Θ) = Π − (Θ) + η − (Θ) ,N − AB ( v, Θ) → N − DAB ( v, Θ) = N − AB ( v − C − (Θ) , Θ) . (20)The new out variable Π + D has canonical commutation relations with C + and N DAB ; in particular,it commutes with N DAB . The in variable Π − D enjoys the same commutation relations with thetransformed canonical variables defined on I − .In terms of the new variables the matching conditions are C + (Θ) = C − (Θ) , Π + D (Θ) = Π − D (Θ) . (21)The generating functional of asymptotic time evolution is a function F [ C − , N − D + AB , Π + D , N D − AB ]of initial coordinates and final momenta. Here the subscripts + and − denote positive andnegative frequencies in u and v .The matching conditions imply C − (Θ A ) = δFδ Π + D (Θ A ) , Π + D (Θ A ) = δFδC − (Θ A ) . (22)This equation is easily solved by F = Z d Θ C − Π + D + f [ N − D + AB , N D − AB ] , (23) F depends also on additional symplectic variables associated to matter, when matter fields are present.They are just additional hard variables conceptually identical to the dressed Bondi news. f is independent of C − , Π + D .This is a key result of our paper. It shows that BMS symmetry does not constrainthe dynamics of the hard degrees of freedom at all, because it tells us nothing aboutthe functional f . To be concrete, suppose that we are given an initial configuration described by the canonicalvariables C − , Π − , N − AB . Naively, we might think that the infinite number of conserved BMScharges, together with the intricate way in which hard and soft modes mix during time evolution,could imprint some information about the initial state on the final configuration of the soft modes C + , Π + . Equation (23) shows that this is a mistake.Once the initial state is rewritten in terms of properly “dressed” variables, the evolutionof the soft modes decouples completely from that of the hard modes. In fact, soft modesevolve trivially. Even more importantly, the evolution of the hard states is the same for anyconfiguration of soft states.The apparent nontrivial mixing of hard and soft modes under time evolution is just anillusion due to a bad choice of coordinates. In other words, the soft hair is a wig. It can bepulled off without affecting the rest of the dynamics. Its “hairs” are conserved because theydecouple, not because they carry any information about the rest of the world. We have proven factorization of the soft dynamics in classical mechanics, but an analogousresult holds also in quantum field theory for the S-matrix, as we will show in the next subsection.
We do not have a complete quantum theory of gravity in asymptotically flat space-times; there-fore, any result that we may hope to obtain must rely on additional assumptions. Ours will bethat there exists a unitary S-matrix that maps quantum fields defined on I − to fields definedon I + .With this assumption, we need not investigate the effect of horizons, which are associatedwith ephemeral intermediate states from the point of view of the S-matrix. We will now describe The memories Π ± (Θ) can be defined in terms of the permanent displacement of sets of massive particlesmoving along appropriate world-lines. Appendix A shows how to dress such quantities in a manner that makesthe factorization of hard dynamics explicit. The soft particles do carry their own information, of course. This information is independent of the hardscattering data, and it cannot be accessed [21–23] on the shorter timescales sufficient for producing and measuringthe hard in and out states. With this caveat in mind, let us proceed toa formal definition of the unitary operator that implements the canonical transformation (20)on states.The classical mapping between in and out variables translates naturally into the Heisenbergpicture of quantum evolution, where operators evolve while states do not. Heisenberg-pictureoperators will carry a subscript H . We need two operators U H , V H that transform the in and out soft variables as U H Π + H U − H = Π + H + η + H , V H Π − H V − H = Π − H + η − H . (24)Since the canonical commutator is[Π + H (Θ) , C + H (Θ ′ )] = − iδ (Θ − Θ ′ ) , (25)the operators U H , V H are U H = e − i R d Θ C + H (Θ) η + H (Θ) , V H = e − i R d Θ C − H (Θ) η − H (Θ) , (26)up to a c-number phase.As we mentioned earlier, the integral over Θ can be replaced with a sum over angularmomenta by expanding η + H (Ω) , C + H (Ω), η − H (Ω) , C − H (Ω) in spherical harmonics. The quantum The “dressing” operator that we shall find is closely related to that used in [24–26] to tame the infraredproblem of QED. While formally unitary, the latter maps any vector belonging to the Fock space of photons, H , into a vector orthogonal to all the vectors in H (see e.g. Eq. 12 of [26]). This is of course incompatible withthe dressing being a well-defined unitary operator in H . We apologize for deviating from our convention of using ± superscripts associated with I ± . The consistentnotation V + H , V − H would be cumbersome because we frequently refer to the inverse, ( V ± ) − . U H Π + H U − H = V H Π − H V − H , U H C + H U − H = V H C − H V − H . (27)On the other hand, the in and out Heisenberg operators are mapped into each other, by ourassumption, by a unitary evolution operator S H :Π + H = S − H Π − H S H , C + H = S − H C − H S H . (28)Eqs. (27,28) imply that the matrix S H U − H V H commutes with Π − H , C − H ; but the latter are canon-ically conjugated operators that commute with all other dynamical variables. Thus, by Schur’slemma, S H U − H V H is proportional to the identity on any irreducible representation of the softcanonical commutator algebra. S H U − H V H acts as a nontrivial unitary matrix only on the Hilbertspace associated to the “hard” operators N AB (plus eventual matter fields).This property allows us to introduce a factorized S-matrix ˆ S H that does not act on the Hilbertspace of soft modes. By their definition, the dressing operators (27) obey U H = S − H V H S H ; itfollows that ˆ S H = S H U − H V H = V − H S H V H → S H = V H ˆ S H V − H . (29)This is almost a factorization formula for the S-matrix.The last step we need is to recall that the usual S-matrix, that is the one that is computedby Feynman diagrams in perturbation theory, is defined in the interaction representation, whereoperators evolve with a “free” S-matrix S , while states evolve with S ≡ S − S H . Operatorsin interaction representation will carry no subscript. A standard choice is that operators inHeisenberg and interaction representation coincide on I − . In the interaction representation, the out operators on I + are defined as O out = S − O in S , so U ≡ S − V S . Finally, equation (29)becomes S = U ˆ SV − , ˆ S ≡ S − ˆ S H . (30)Readers familiar with the problem of infrared divergences in quantum field theory will recog-nize the similarity between the factorization formula (30), and the Block-Nordsieck factorizationthat is the key to solving the “infrared catastrophe.” This is not a coincidence: Eq. (30) is theextreme infrared limit of the formulas used in [11] to factorize soft U (1) and BMS hair. Thefinite-frequency counterparts of U, V were introduced in 1965 for QED [24]. They were furtherstudied in [25, 26]. As usual in theoretical physics, they are known by the name of the last11uthors that discovered them. Factorization of Large U (1) Soft Hair Dynamics
Classical and quantum factorization formulas in this case are completely analogous to the BMScase after the obvious substitution C + → φ + , C − → φ − . (31)A field on I + carrying no U (1) charge needs no dressing, since it already commutes with thecharge Q + [ f ], while a field Ψ of charge q commutes with Q + [ f ] after the dressingΨ( u, Θ) → e − iqφ + (Θ) Ψ( u, Θ) . (32) We defined dressed variables somewhat formally through the canonical transformations (20),(31), or their quantum analogues, the dressing operators
U, V . In this section, we will give aphysical interpretation of this operation: it removes the contribution to the soft charge com-ing from the long-range field of the hard particles. We do this by explicitly computing thiscontribution and showing that it equals minus the hard charge, up to a constant. U (1) Classical Dressed States
In the metric (2) the angular components of Maxwell’s equations are − ∂ r ( √ γγ AB F uB ) − ∂ u ( √ γγ AB F rB ) + ∂ r ( √ γγ AB F rB ) + 1 r ∂ C ( √ γγ CD γ AB F DB ) = √ γγ AB e J B . (33)So, by integrating in r and using F rA = O ( r − ) [1] we arrive at a particular solution: F Au ( r, u, Θ) = − Z r ds (cid:2) e J A ( s, u, Θ) + ∂ u F Ar ( r, u, Θ) (cid:3) + D C X CA ( r, u, Θ) . (34)The function X AB is antisymmetric. Using the explicit form of the soft charge Q + s [ f ] givenin Eq. (3) we see that the term proportional to X AB vanishes after the angular integration; The relation between various formulations of the QED dressing factors was recently revisited in [27]. Q + s [ f ] = − Z ∞ ds Z d Θ √ γ f (Θ) h e D A F Ar ( s, u, Θ) (cid:12)(cid:12)(cid:12) u =+ ∞ u = −∞ + Z + ∞−∞ du D A J A ( s, u, Θ) i . (35)The first term in brackets vanishes at each finite value of s , because at time-like infinity i ± allfield strengths vanish in massless QED.In the metric (2), the current conservation equation is − ∂ r √ γr J u − ∂ u √ γr J r + ∂ r √ γr J r + √ γD A J A = 0 . (36)The field strengths F ur , F Ar are O ( r − ) at large r [1]. The r component of Maxwell’s equationsis ∂ r r F ur + D A F Ar = r e J r . (37)Substituting the asymptotic behavior of the field strengths into Eq. (37) we find lim r →∞ r J r = 0.So we have lim r →∞ r J u ( r, u, Θ) = Z ∞ ds h D A J A ( s, u, Θ) − ∂ u s J r ( s, u, Θ) i . (38)At all finite r , the current component r J r vanishes at u = ±∞ , because no current escapesfrom i ± . So the definition of Q + s gives finally Q + s [ f ] = − Z + ∞−∞ du Z d Θ √ γf (Θ) j u ( u, Θ) = − Q + h . (39)This last equation expresses the equality that we have been looking for:“hard charge = − gravitational memory of the dressing,” (40)if the ambiguity in the dressing operator is resolved by the choice (26).Notice that in Eqs. (35,39), only the zero frequency mode of currents contribute (becauseof the integral in u ). So, the definition of the dressed state is not unique. For instance, it canbe changed by adding to the F Au defined in Eq. (34) an arbitrary solution of the homogeneousMaxwell equations, F hom Au , as long as its Fourier tranform ˜ F hom Au ( ω ) = R + ∞−∞ du exp( iωu ) F hom Au vanishes at ω = 0. One can also make the total charge nonzero by adding to the soft chargeonly photons with frequency below any arbitrarily small infrared cutoff ω > .2 BMS Classical Dressing The equations of motion of general relativity are nonlinear, so one cannot simply define dressedstates by subtracting a particular solution of Einstein’s equations from the bare fields. Moreover,the definition of finite-time or finite-radius fields is inherently ambiguous. Therefore the methodsof the previous section cannot be used. The alternative method we employ here is less directlyconnected with the scattering problem, but it has two merits: 1) it only uses asymptotic data;2) it makes clear that the cancellation of the hard charge can be achieved by changing only softmodes.First of all, change the Fourier transform of the Bondi news as follows˜ N AB ( ω ) → ˜ N AB ( ω ) + χ ( ω )[ − D A D B X (Θ) + γ AB D X (Θ)] , ˜ N AB = Z due iωu N AB ( u ) , (41)where χ ( ω ) is a step function obeying χ ( ω ) = 1 for | ω | < ω and χ ( ω ) = 0 for | ω | > ω . The IRcutoff ω can be arbitrarily small. Write the BMS charge in terms of ˜ NQ + = Q + h + Q + s , Q h [ f ] = 14 πG Z dω π Z d Θ √ γ f (Θ) ˜ N AB ˜ N AB + Q + M ,Q + s [ f ] = − πG lim ω → Z d Θ √ γ f (Θ) D A D B ˜ N AB ( ω ) ,Q + M = 14 π lim r →∞ Z du d Θ √ γ f (Θ) r T Muu . (42)Under the transformation (41) the charges change as follows Q + s [ f ] → Q + s [ f ] + 116 πG Z d Θ √ γf (Θ) D X (Θ) , Q + h [ f ] → Q + h [ f ] + O ( ω ) . (43)For ω → Q + h is negligible. In the same limit, the hard charge can be canceledby satisfying the equation D ( D + 2) X = − Z dω π ˜ N AB ˜ N AB − πGQ + M . (44)Notice that the Bondi news is unchanged for frequencies larger than ω .14 cknowledgments It is a pleasure to thank V. Chandrasekharan, E. Flanagan, I. Halpern, S. Leichenauer, A. Stro-minger, and A. Wall for discussions. R.B. was supported in part by the Berkeley Center forTheoretical Physics, by the National Science Foundation (award numbers PHY-1521446, PHY-1316783), by FQXi, and by the US Department of Energy under contract DE-AC02-05CH11231.M.P. was supported in part by NSF grants PHY-1316452, PHY-1620039. M.P. thanks theGalileo Galilei Institute for Theoretical Physics (GGI) for hospitality and INFN for partial sup-port during the completion of this work, within the program “New Developments in AdS /CFT Holography.”
A Massive Particles as Gravitational Memory Detectors
There is one more class of observables that has played a prominent role in the study of BMSsymmetry. These are the gravitational memories measured by displacement of massive particleseither moving along geodesics or kept at fixed radius and fixed angular coordinates [28]. Thesequantities define a particular, concrete procedure to measure the memories Π ± (Θ) so they cannotchange the conclusions reached in the paper. Once properly dressed, they too will manifestlyfactor out of the nonzero-frequency dynamics. This appendix shows explicitly how to dressthese variables, but we begin by defining them. Gravitational Memories as Scattering Data
At large r the metric in retarded coordinates is given in Eq. (13), while in advanced coordinatesit is given in Eq. (18). At large but finite r the antipodal map between retarded angularcoordinates Θ A and advanced angular coordinates θ A isΘ A = θ A + 1 r f A ( θ, v ) + O ( r − ) . (A.1)Consider next a bunch of detectors at fixed positions r , Θ i and retarded time u . As in [28]we consider nearby detectors so that the distance L ij between any two of them at equal time isapproximately L ij = [ r γ AB + rC AB ](Θ Ai − Θ Aj )(Θ Bi − Θ Bj ) . (A.2)15s C AB evolves from u = −∞ to u = + ∞ the detectors’ relative position changes as [28] L ij | u =+ ∞ − L ij | u = −∞ = 12 L ij r ( C AB | u =+ ∞ − C AB | u = −∞ )(Θ Ai − Θ Aj )(Θ Bi − Θ Bj ) . (A.3)This equation does not give any new information on gravitational scattering, since it simplydetermines the final positions of the detectors, L ij | u =+ ∞ , in terms of the change in C AB andthe initial positions L ij | u = −∞ . In other words Eq. (A.3) solves the scattering problem for aparticular type of massive objects, i.e. the detectors.Notice that an analogous equation can be written in terms of the advanced coordinates andthe metric components C − AB as L ij | v =+ ∞ − L ij | v = −∞ = 12 L ij r ( C − AB | v =+ ∞ − C − AB | v = −∞ )( θ Ai − θ Aj )( θ Bi − θ Bj ) . (A.4)Even though the LHS of Eqs. (A.3,A.4) is the same, we cannot use those equations to expressthe change in C − AB in terms of the change of C AB , because we do not know the functions f A inEq. (A.1). The best we can do is to use the two equations to constrain f A ( θ, v ). Specifically,2 L ij γ AC ∆ f CB j = C AB | u =+ ∞ − C AB | u = −∞ − C − AB | v =+ ∞ + C − AB | v = −∞ , (A.5)where ∆ f AB j = ∂ B f A ( θ j , + ∞ ) − ∂ B f A ( θ j , −∞ ). Dressed Gravitational Memories
The effect of soft BMS hair on the dynamics of gravitational memories can be factored outby defining appropriate variables that are insensitive to the IR fields. This must be true alsofor the variables defined in this appendix, which are just a concrete realization of gravitationalmemories. Dressed Bondi news were defined in Eq. (20). Equation (A.3) suggests how to definethe “dressed memory” observables that we defined here: L Dij ≡ L ij − L ij rC AB (Θ Ai − Θ Aj )(Θ Bi − Θ Bj ) . (A.6)These variables remain constant under time evolution so L Dij | u =+ ∞ = L Dij | u = −∞ so they areobviously independent of the soft graviton state.16 eferences [1] T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of Massless QED,”JHEP , 112 (2014) doi:10.1007/JHEP10(2014)112 [arXiv:1407.3789 [hep-th]].[2] M. Campiglia and A. Laddha, “Asymptotic symmetries of QED and Weinbergs soft photontheorem,” JHEP , 115 (2015) doi:10.1007/JHEP07(2015)115 [arXiv:1505.05346 [hep-th]].[3] D. 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